How to solve a system of equations of linear type

For a complete understanding of how to solve a system of equations, you should consider what it is. As is clear from the term itself, a "system" is a collection of several equations related to each other. There are systems of algebraic and differential equations. In this article we will pay attention to how to solve a system of equations of the first type.
By definition, an equation is called algebraic, In which only simple mathematical operations are performed on the variables, i.e. Addition, division, subtraction, multiplication, exponentiation, and finding the root. An algorithm for solving an equation of this type reduces to finding a equivalent, but a simpler construction, by transforming it.
Systems of algebraic equations are subdivided into linear and nonlinear.
System of linear equations (also widely used abbreviation SLAE) differs from the system of nonlinear equations in that unknown variables here are in the first degree. The general form of SLAE in matrix entries is as follows: Ax = b, where A is the set of known coefficients, x are variables, and b is the set of known free terms.

There are many ways of how to solve a system of equations of this type, they Are subdivided into direct and iterative methods. Direct methods allow us to find the values of variables for a certain number of mathematical transformations, and iterative algorithms use the algorithm of successive approximation and refinement.

Let us analyze by an example how to solve a system of linear equations using a direct method of finding the value of variables. Direct methods include the methods of Gauss, Jordan-Gauss, Cramer, sweeps and some others. One of the simplest can be called the method of Cramer, usually it is with him in the curriculum begins acquaintance with the matrix. This method is designed to solve square SLAU, i.e. Such systems, in which the number of equations is equal to the number of unknown variables in a row. Also, in order to solve the system of equations by the Cramer method, it is necessary to make sure that the free terms are not zeros (this is a necessary condition).

The solution algorithm is as follows: a matrix 1 is composed of known coefficients of the a-system and its main determinant Δx is found. The determinant is found by subtracting the product of the elements of the secondary diagonal from the product of the elements The main one.

Next, a matrix 2 is compiled, where the values of the free elements b are substituted in the first column, similar to the previous example, the determinant Δx _{1 is found} .

We compose the matrix 3, we substitute the values of the free coefficients already in the second column, we find the determinant of the matrix Δx ₂ . And so on, until we compute the determinant of that matrix, where the coefficients b are in the last column.

To find the value of a particular variable, the determinants obtained by substituting the free coefficients must be divided into the main determinant, i.e. X ₁ = Δх ₁ / Δх, х ₂ = Δх ₂ / Δх, etc.
If you have any questions on how to solve the system of equations in one way or another, I recommend that you refer to the reference and educational material, which details all the main steps.